The Larson-Sweedler theorem for Hopf categories

The classical theorem of Larson and Sweedler (1969) tells that a finite dimensional bialgebra is Hopf if and only if it has a non-singular integral, and that moreover this integral is unique (up to scalar multiplication). As a consequence, an algebra that satisfies these equivalent conditions is always Frobenius. It was one of the first results relating Hopf and Frobenius structures on the same underlying algebra.

Hopf V-categories (where V is a braided monoidal category) are a recent generalization of Hopf algebras. A Hopf V-category with one object is just a usual Hopf algebra in V, and a Hopf Set-category is nothing else than a groupoid. In this way, Hopf V-categories can be seen as a “many-object version” of Hopf categories and lead naturally to examples of weak (multiplier) Hopf algebras. 

The first aim of this talk is to explain the definition of a Hopf category and their link with weak Hopf algebras and “oplax Hopf algebras”. Secondly we will develop integral theory for Hopf categories and show the Larson-Sweedler theorem for Hopf categories.

This is joint work with Mitchell Buckley, Timmy Fieremans and Christina Vasilakopoulou.